Thread: Proving...

The question asks:

Prove that if (v1, ... , vn) is a basis of V, then so is(v1, v2-v1, ... , vn - vn-1).

I made the initial assumption that if V remained the same number that by subtracting v1, youd always end up with v1, and so on. But obviously I came to the conclusion that this would have to be wrong, because the set of (v1, .. , vn) could represent any numbers. So I have no idea how to tackle this one, any help would be greatly appreciated. Thanks.

If are such that then for all . Now suppose there are scalars such that where then we get where then , and so ... . Which means they're l.i. and therefore a basis.

Thank you very much Jose, but I am unsure as to whether I can approach the question like this. It would almost appear as if that is proving that they are a basis rather than specifically a basis for V. Would it be possible to get a second opinion on this?

Any set of 'n' independent vectors which belong to a n-dimensional vector space,V, will be a basis of V.
(This is a std theorm and can be found in most of the texts on linear algebra)

Originally Posted by aman_cc

Any set of 'n' independent vectors which belong to a n-dimensional vector space,V, will be a basis of V.
(This is a std theorm and can be found in most of the texts on linear algebra)

Yes I understand that it is a standard theorem, just as we can say X²=4 when x=2. But that is not what I am asking, I am asking how to PROVE this theorem. That is a completely different method. However I appreciate your input.

Originally Posted by GreenDay14

Thank you very much Jose, but I am unsure as to whether I can approach the question like this. It would almost appear as if that is proving that they are a basis rather than specifically a basis for V. Would it be possible to get a second opinion on this?

What is your definition of basis?

It would almost appear as if that is proving that they are a basis rather than specifically a basis for V.

You are given that the first n vectors form a basis for V so you know that V has dimension n. You are given another n vectors in V. To show that they are a basis for V you need only show that they span V or that they are independent.